Riemann curvature-stretching coupling in dynamo torus laboratory and in UHF twisted plasma loops

Abstract

A plasma loop twisted Riemannian model is applied to torus dynamos twisted flows it leading to a slow dynamo such as in Moebius strip dynamo, recently considered by Shukurov, Stepanov and Sokoloff [Phys. Rev. \textbf{E 78},025301,(2008)] to modelling Perm dynamo torus in liquid sodium. Since diffusion and advection (stretching), are competing effects for dynamo action, plasma resistivity term is shown to be proportional to loops Riemann curvature (folding). Shukurov et al, also showed that based on Ponomarenko dynamo, a broader torus channel produces a better dynamo. These results agree with Schekochihin et al [Phys Rev \textbf{E} (2002)] where random filamentary magnetic fields are strengthen by curvature. Analysis of spectrum of chaotic fast dynamos, shows that Riemann curvature acts as a damping, since growth magnetic field rate is inversely proportional to Riemann curvature. Comparison with general relativistic MHD dynamo equation, shows that the Ricci tensor, which is a contraction of the Riemann tensor also appears in the diffusion term. Curvature of plasma loop is ${{R^{1}}_{212}}|_{\textbf{Plasma}}\approx{5.6{\times}10^{-19}m^{-2}}$, while for Perm torus is certainly higher. Thus slow dynamos are favoured in dynamo laboratories rather than in plasma loops. It is shown that the curvature-stretching flux rope dynamo coupling energy, coincides with the minimum twist energy ${\epsilon}_{\textbf{twist}}\approx{10^{30}TeV}$ stored in flux ropes. Torus flux tubes around black-holes remain in the order of $2MeV$ and GBR are around $10^{52}TeV$. Since the ${R^{1}}_{212}$ is negative, inflexionary flux tubes fast dynamos may be responsible for this CME mechanism in UHF plasma loops.